The problem you pose is notoriously difficult, I think. I was actually thinking about another problem, specifically, to show that for any n >= 0, there exists a circle whose interior includes precisely n integer points. An outline of my proof: Let O be a point in the plane which is not equidistant from two integer points. Then any circle centered at O passes through at most one integer point. Take the sequence {C_n | n = 0, 1, 2...} of circles centered at O and passing through an integer point, ordered by increasing radius. We can show that C_n has exactly n points in its interior. Now we need only show that O exists. Certainly any O = (x, y) with mutually transcendental coordinates works, but I was hoping to find point O with explicit coordinates. This got me wondering exactly which points O were candidates, specifically, exactly which points of the plane are not equidistant from two integer points. ----- Original Message ----- From: asimovd@aol.com To: math-fun Sent: Thursday, January 22, 2004 1:54 AM Subject: Re: [math-fun] Simple Question I think David Wilson asks: << Is every rational point on the plane equidistant from two integer points?

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Here's a question in this vein that I think is unsolved: In the plane, given a unit square, is there any point whose distance to each corner is rational? Dan -- NOTE: Please direct any responses to <asimov@msri.org>, since I may not keep this AOL e-address much longer -- thanks. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun